3.2.93 \(\int (a+b \text {sech}^2(c+d x))^{5/2} \, dx\) [193]
Optimal. Leaf size=170 \[ \frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b-b \tanh ^2(c+d x)}}\right )}{8 d}+\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+b-b \tanh ^2(c+d x)}}\right )}{d}+\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a+b-b \tanh ^2(c+d x)}}{8 d}+\frac {b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d} \]
[Out]
a^(5/2)*arctanh(a^(1/2)*tanh(d*x+c)/(a+b-b*tanh(d*x+c)^2)^(1/2))/d+1/8*(15*a^2+10*a*b+3*b^2)*arctan(b^(1/2)*ta
nh(d*x+c)/(a+b-b*tanh(d*x+c)^2)^(1/2))*b^(1/2)/d+1/8*b*(7*a+3*b)*(a+b-b*tanh(d*x+c)^2)^(1/2)*tanh(d*x+c)/d+1/4
*b*tanh(d*x+c)*(a+b-b*tanh(d*x+c)^2)^(3/2)/d
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4213, 427, 542,
537, 223, 209, 385, 212} \begin {gather*} \frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-b \tanh ^2(c+d x)+b}}\right )}{d}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a-b \tanh ^2(c+d x)+b}}\right )}{8 d}+\frac {b \tanh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )^{3/2}}{4 d}+\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a-b \tanh ^2(c+d x)+b}}{8 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*Sech[c + d*x]^2)^(5/2),x]
[Out]
(Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b - b*Tanh[c + d*x]^2]])/(8*d) + (a
^(5/2)*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + b - b*Tanh[c + d*x]^2]])/d + (b*(7*a + 3*b)*Tanh[c + d*x]*Sqrt
[a + b - b*Tanh[c + d*x]^2])/(8*d) + (b*Tanh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^(3/2))/(4*d)
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 223
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 427
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]
Rule 537
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]
Rule 542
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]
Rule 4213
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^{5/2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b-b x^2\right )^{5/2}}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}-\frac {\text {Subst}\left (\int \frac {\sqrt {a+b-b x^2} \left ((a+b) (b-4 (a+b))+b (7 a+3 b) x^2\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a+b-b \tanh ^2(c+d x)}}{8 d}+\frac {b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}+\frac {\text {Subst}\left (\int \frac {(a+b) \left (8 a^2+7 a b+3 b^2\right )-b \left (15 a^2+10 a b+3 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a+b-b \tanh ^2(c+d x)}}{8 d}+\frac {b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b-b x^2}} \, dx,x,\tanh (c+d x)\right )}{d}+\frac {\left (b \left (15 a^2+10 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a+b-b \tanh ^2(c+d x)}}{8 d}+\frac {b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tanh (c+d x)}{\sqrt {a+b-b \tanh ^2(c+d x)}}\right )}{d}+\frac {\left (b \left (15 a^2+10 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\tanh (c+d x)}{\sqrt {a+b-b \tanh ^2(c+d x)}}\right )}{8 d}\\ &=\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b-b \tanh ^2(c+d x)}}\right )}{8 d}+\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+b-b \tanh ^2(c+d x)}}\right )}{d}+\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a+b-b \tanh ^2(c+d x)}}{8 d}+\frac {b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 8.22, size = 280, normalized size = 1.65 \begin {gather*} \frac {\left (\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a+b+a \sinh ^2(c+d x)}}\right )+8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b+a \sinh ^2(c+d x)}}\right )\right ) \cosh ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^{5/2}}{\sqrt {2} d (a+2 b+a \cosh (2 c+2 d x))^{5/2}}+\frac {\cosh ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^{5/2} \left (\frac {b^2 \text {sech}(c) \text {sech}^4(c+d x) \sinh (d x)}{d}+\frac {3 \text {sech}(c) \text {sech}^2(c+d x) \left (3 a b \sinh (d x)+b^2 \sinh (d x)\right )}{2 d}+\frac {3 b (3 a+b) \text {sech}(c+d x) \tanh (c)}{2 d}+\frac {b^2 \text {sech}^3(c+d x) \tanh (c)}{d}\right )}{(a+2 b+a \cosh (2 c+2 d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sech[c + d*x]^2)^(5/2),x]
[Out]
((Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a + b + a*Sinh[c + d*x]^2]] + 8*a^(5/2
)*ArcTanh[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b + a*Sinh[c + d*x]^2]])*Cosh[c + d*x]^5*(a + b*Sech[c + d*x]^2)^(5
/2))/(Sqrt[2]*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^(5/2)) + (Cosh[c + d*x]^5*(a + b*Sech[c + d*x]^2)^(5/2)*((b^2*
Sech[c]*Sech[c + d*x]^4*Sinh[d*x])/d + (3*Sech[c]*Sech[c + d*x]^2*(3*a*b*Sinh[d*x] + b^2*Sinh[d*x]))/(2*d) + (
3*b*(3*a + b)*Sech[c + d*x]*Tanh[c])/(2*d) + (b^2*Sech[c + d*x]^3*Tanh[c])/d))/(a + 2*b + a*Cosh[2*c + 2*d*x])
^2
________________________________________________________________________________________
Maple [F]
time = 2.83, size = 0, normalized size = 0.00 \[\int \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sech(d*x+c)^2)^(5/2),x)
[Out]
int((a+b*sech(d*x+c)^2)^(5/2),x)
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate((b*sech(d*x + c)^2 + a)^(5/2), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2712 vs.
\(2 (148) = 296\).
time = 0.90, size = 12452, normalized size = 73.25 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^(5/2),x, algorithm="fricas")
[Out]
[1/16*(4*(a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sinh(d*x + c)^8 + 4*a^2*cosh(d*x + c
)^6 + 4*(7*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^6 + 6*a^2*cosh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^3 + 3*a
^2*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 + 30*a^2*cosh(d*x + c)^2 + 3*a^2)*sinh(d*x + c)^
4 + 4*a^2*cosh(d*x + c)^2 + 8*(7*a^2*cosh(d*x + c)^5 + 10*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x
+ c)^3 + 4*(7*a^2*cosh(d*x + c)^6 + 15*a^2*cosh(d*x + c)^4 + 9*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^2 + a^
2 + 8*(a^2*cosh(d*x + c)^7 + 3*a^2*cosh(d*x + c)^5 + 3*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c))
*sqrt(a)*log((a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 - 2*(a*b^2
- b^3)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 - a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x +
c)^3 - 3*(a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a*b^2*
cosh(d*x + c)^4 + a^3 + 4*a^2*b + 9*a*b^2 - 30*(a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a*b^2*co
sh(d*x + c)^5 - 10*(a*b^2 - b^3)*cosh(d*x + c)^3 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
a^3 + 2*(a^3 + 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 - 15*(a*b^2 - b^3)*cosh(d*x + c)^4 + a^3
+ 3*a^2*b + 3*(a^3 + 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6 + 6*b
^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 - 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 - b^
2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 + 4*a*b)*cosh(d*x
+ c)^2 + (15*b^2*cosh(d*x + c)^4 - 18*b^2*cosh(d*x + c)^2 - a^2 - 4*a*b)*sinh(d*x + c)^2 - a^2 + 2*(3*b^2*cosh
(d*x + c)^5 - 6*b^2*cosh(d*x + c)^3 - (a^2 + 4*a*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*cosh(d*x + c
)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(2
*a*b^2*cosh(d*x + c)^7 - 3*(a*b^2 - b^3)*cosh(d*x + c)^5 + (a^3 + 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^3 + (a^3 +
3*a^2*b)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh(d*x + c)^4
*sinh(d*x + c)^2 + 20*cosh(d*x + c)^3*sinh(d*x + c)^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh(d*x + c)*s
inh(d*x + c)^5 + sinh(d*x + c)^6)) + ((15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^8 + 8*(15*a^2 + 10*a*b + 3*b^2)*
cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^2 + 10*a*b + 3*b^2)*sinh(d*x + c)^8 + 4*(15*a^2 + 10*a*b + 3*b^2)*cosh(d
*x + c)^6 + 4*(7*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^2 + 15*a^2 + 10*a*b + 3*b^2)*sinh(d*x + c)^6 + 8*(7*(
15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^3 + 3*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(15*
a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^4 + 30*(15*a^2 + 10*a*b
+ 3*b^2)*cosh(d*x + c)^2 + 45*a^2 + 30*a*b + 9*b^2)*sinh(d*x + c)^4 + 8*(7*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x
+ c)^5 + 10*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^3 + 3*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x +
c)^3 + 4*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^2 + 4*(7*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^6 + 15*(15*a
^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^4 + 9*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^2 + 15*a^2 + 10*a*b + 3*b^2)*
sinh(d*x + c)^2 + 15*a^2 + 10*a*b + 3*b^2 + 8*((15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^7 + 3*(15*a^2 + 10*a*b
+ 3*b^2)*cosh(d*x + c)^5 + 3*(15*a^2 + 10*a*b + 3*b^2)*cosh(d*x + c)^3 + (15*a^2 + 10*a*b + 3*b^2)*cosh(d*x +
c))*sinh(d*x + c))*sqrt(-b)*log(-((a - b)*cosh(d*x + c)^4 + 4*(a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - b)*
sinh(d*x + c)^4 + 2*(a + 3*b)*cosh(d*x + c)^2 + 2*(3*(a - b)*cosh(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2 - 2*sq
rt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-b)*sqrt((a*cosh(d*x + c)^2
+ a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*((a -
b)*cosh(d*x + c)^3 + (a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + a - b)/(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(
d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c
)^3 + cosh(d*x + c))*sinh(d*x + c) + 1)) + 4*(a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*
sinh(d*x + c)^8 + 4*a^2*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^6 + 6*a^2*cosh(d*x + c
)^4 + 8*(7*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 + 30*a^2*cos
h(d*x + c)^2 + 3*a^2)*sinh(d*x + c)^4 + 4*a^2*cosh(d*x + c)^2 + 8*(7*a^2*cosh(d*x + c)^5 + 10*a^2*cosh(d*x + c
)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^2*cosh(d*x + c)^6 + 15*a^2*cosh(d*x + c)^4 + 9*a^2*cosh(d*
x + c)^2 + a^2)*sinh(d*x + c)^2 + a^2 + 8*(a^2*cosh(d*x + c)^7 + 3*a^2*cosh(d*x + c)^5 + 3*a^2*cosh(d*x + c)^3
+ a^2*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*log(-(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*s
inh(d*x + c)^4 + 2*(a + b)*cosh(d*x + c)^2 + 2*...
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)**2)**(5/2),x)
[Out]
Integral((a + b*sech(c + d*x)**2)**(5/2), x)
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.53Error: Bad Argument Type
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b/cosh(c + d*x)^2)^(5/2),x)
[Out]
int((a + b/cosh(c + d*x)^2)^(5/2), x)
________________________________________________________________________________________